Properties

Label 144.2.x.c
Level $144$
Weight $2$
Character orbit 144.x
Analytic conductor $1.150$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 144.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} -2 \zeta_{12} q^{4} + ( 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{6} + ( 2 + \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{7} + ( -2 + 2 \zeta_{12}^{3} ) q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{2} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{3} -2 \zeta_{12} q^{4} + ( 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{5} + ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{6} + ( 2 + \zeta_{12} - \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{7} + ( -2 + 2 \zeta_{12}^{3} ) q^{8} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( 2 - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{10} + ( 3 - \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} + ( 2 - 4 \zeta_{12}^{2} ) q^{12} + ( 3 - 3 \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{13} + ( -2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{14} + ( 4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{15} + 4 \zeta_{12}^{2} q^{16} + ( -4 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{17} + ( 3 + 3 \zeta_{12}^{3} ) q^{18} + ( -2 - \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{19} + ( -4 + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{20} + ( 1 + 3 \zeta_{12} + \zeta_{12}^{2} ) q^{21} + ( -3 - \zeta_{12} - 3 \zeta_{12}^{2} ) q^{22} + ( -1 - 3 \zeta_{12} - \zeta_{12}^{2} ) q^{23} + ( -4 - 2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{24} + ( -8 + 3 \zeta_{12} + 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{25} + ( -2 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{26} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + ( -2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{28} + ( 2 + \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{29} + ( -4 + 6 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{30} + ( -4 + 2 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{31} + ( 4 + 4 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{32} + ( 5 + 5 \zeta_{12} - 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{33} + ( -2 - 3 \zeta_{12} + 5 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{34} + ( 2 - 2 \zeta_{12}^{3} ) q^{35} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{36} + ( 2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{37} + ( \zeta_{12} + 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{38} + ( -5 + 2 \zeta_{12} - 2 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{39} + ( 4 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{40} -3 \zeta_{12} q^{41} + ( 4 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{42} + ( 3 + 2 \zeta_{12} - 5 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{43} + ( -4 - 6 \zeta_{12} + 6 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{44} + ( -6 + 6 \zeta_{12} ) q^{45} + ( -4 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{46} + ( 5 \zeta_{12} - \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{47} + ( -4 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{48} + ( -3 + 2 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{49} + ( 7 - 7 \zeta_{12} + \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{50} + ( -4 \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{51} + ( 8 - 6 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{52} + ( -2 - 4 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{53} + ( -6 + 3 \zeta_{12} + 3 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{54} + ( -2 + 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{55} + ( -4 + 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{56} + ( -3 - 3 \zeta_{12} ) q^{57} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{58} + ( 5 + 5 \zeta_{12} - 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{59} + ( 8 - 8 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{60} + ( 6 + 6 \zeta_{12} ) q^{61} + ( 6 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{62} + ( -3 + 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{63} -8 \zeta_{12}^{3} q^{64} + ( -6 \zeta_{12} + 14 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{65} + ( 1 - 2 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{66} + ( -2 + 2 \zeta_{12} - 3 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{67} + ( 2 + 8 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{68} + ( 3 - 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{69} -4 \zeta_{12}^{2} q^{70} + ( 4 - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{71} + ( 6 - 6 \zeta_{12} - 6 \zeta_{12}^{2} ) q^{72} + ( 1 - 2 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{73} + ( 4 + 6 \zeta_{12} - 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{74} + ( 3 - 12 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{75} + ( 4 + 4 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{76} + ( 3 - 3 \zeta_{12} - 7 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{77} + ( -2 \zeta_{12} + 12 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{78} -12 \zeta_{12}^{2} q^{79} + ( -8 + 8 \zeta_{12} + 8 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{80} -9 \zeta_{12}^{2} q^{81} + ( -3 + 3 \zeta_{12}^{3} ) q^{82} + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{83} + ( -2 \zeta_{12} - 6 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{84} + ( -2 - 2 \zeta_{12} - 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{85} + ( -3 - 7 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{86} + ( -3 + 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{87} + ( 6 \zeta_{12} + 2 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{88} + 2 \zeta_{12}^{3} q^{89} + ( 6 - 6 \zeta_{12} + 6 \zeta_{12}^{2} ) q^{90} + ( 4 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{91} + ( 2 \zeta_{12} + 6 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{92} + ( 6 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{93} + ( 4 + 4 \zeta_{12} + 6 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{94} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{95} + ( -4 + 8 \zeta_{12} + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{96} + ( -\zeta_{12} - 10 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{97} + ( 5 - 4 \zeta_{12} - 4 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{98} + ( -3 + 9 \zeta_{12} + 9 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 4q^{5} + 6q^{6} + 6q^{7} - 8q^{8} - 6q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 4q^{5} + 6q^{6} + 6q^{7} - 8q^{8} - 6q^{9} + 8q^{11} + 14q^{13} - 4q^{14} + 12q^{15} + 8q^{16} - 16q^{17} + 12q^{18} - 6q^{19} - 8q^{20} + 6q^{21} - 18q^{22} - 6q^{23} - 12q^{24} - 24q^{25} - 8q^{26} - 8q^{28} + 6q^{29} - 12q^{30} - 8q^{31} + 8q^{32} + 12q^{33} + 2q^{34} + 8q^{35} + 12q^{37} + 6q^{38} - 24q^{39} + 8q^{40} + 12q^{42} + 2q^{43} - 4q^{44} - 24q^{45} - 12q^{46} - 2q^{47} - 6q^{49} + 30q^{50} - 6q^{51} + 28q^{52} - 16q^{53} - 18q^{54} - 8q^{56} - 12q^{57} + 12q^{59} + 24q^{60} + 24q^{61} + 16q^{62} + 28q^{65} - 14q^{67} + 12q^{68} - 8q^{70} + 12q^{72} + 12q^{74} + 18q^{75} + 12q^{76} - 2q^{77} + 24q^{78} - 24q^{79} - 16q^{80} - 18q^{81} - 12q^{82} - 2q^{83} - 12q^{84} - 16q^{85} - 18q^{86} - 6q^{87} + 4q^{88} + 36q^{90} + 20q^{91} + 12q^{92} + 24q^{93} + 28q^{94} - 20q^{97} + 12q^{98} + 6q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(-1 + \zeta_{12}^{2}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
−1.36603 0.366025i −0.866025 1.50000i 1.73205 + 1.00000i 1.00000 + 3.73205i 0.633975 + 2.36603i 0.633975 0.366025i −2.00000 2.00000i −1.50000 + 2.59808i 5.46410i
61.1 0.366025 + 1.36603i 0.866025 1.50000i −1.73205 + 1.00000i 1.00000 + 0.267949i 2.36603 + 0.633975i 2.36603 + 1.36603i −2.00000 2.00000i −1.50000 2.59808i 1.46410i
85.1 0.366025 1.36603i 0.866025 + 1.50000i −1.73205 1.00000i 1.00000 0.267949i 2.36603 0.633975i 2.36603 1.36603i −2.00000 + 2.00000i −1.50000 + 2.59808i 1.46410i
133.1 −1.36603 + 0.366025i −0.866025 + 1.50000i 1.73205 1.00000i 1.00000 3.73205i 0.633975 2.36603i 0.633975 + 0.366025i −2.00000 + 2.00000i −1.50000 2.59808i 5.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
144.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.x.c yes 4
3.b odd 2 1 432.2.y.b 4
4.b odd 2 1 576.2.bb.c 4
9.c even 3 1 144.2.x.b 4
9.d odd 6 1 432.2.y.c 4
12.b even 2 1 1728.2.bc.a 4
16.e even 4 1 144.2.x.b 4
16.f odd 4 1 576.2.bb.d 4
36.f odd 6 1 576.2.bb.d 4
36.h even 6 1 1728.2.bc.d 4
48.i odd 4 1 432.2.y.c 4
48.k even 4 1 1728.2.bc.d 4
144.u even 12 1 1728.2.bc.a 4
144.v odd 12 1 576.2.bb.c 4
144.w odd 12 1 432.2.y.b 4
144.x even 12 1 inner 144.2.x.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.2.x.b 4 9.c even 3 1
144.2.x.b 4 16.e even 4 1
144.2.x.c yes 4 1.a even 1 1 trivial
144.2.x.c yes 4 144.x even 12 1 inner
432.2.y.b 4 3.b odd 2 1
432.2.y.b 4 144.w odd 12 1
432.2.y.c 4 9.d odd 6 1
432.2.y.c 4 48.i odd 4 1
576.2.bb.c 4 4.b odd 2 1
576.2.bb.c 4 144.v odd 12 1
576.2.bb.d 4 16.f odd 4 1
576.2.bb.d 4 36.f odd 6 1
1728.2.bc.a 4 12.b even 2 1
1728.2.bc.a 4 144.u even 12 1
1728.2.bc.d 4 36.h even 6 1
1728.2.bc.d 4 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 4 T_{5}^{3} + 20 T_{5}^{2} - 32 T_{5} + 16 \) acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( 16 - 32 T + 20 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 4 - 12 T + 14 T^{2} - 6 T^{3} + T^{4} \)
$11$ \( 169 - 130 T + 41 T^{2} - 8 T^{3} + T^{4} \)
$13$ \( 484 - 220 T + 74 T^{2} - 14 T^{3} + T^{4} \)
$17$ \( ( 13 + 8 T + T^{2} )^{2} \)
$19$ \( 9 + 18 T + 18 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( 36 - 36 T + 6 T^{2} + 6 T^{3} + T^{4} \)
$29$ \( 36 - 36 T + 18 T^{2} - 6 T^{3} + T^{4} \)
$31$ \( 16 + 32 T + 60 T^{2} + 8 T^{3} + T^{4} \)
$37$ \( 144 - 144 T + 72 T^{2} - 12 T^{3} + T^{4} \)
$41$ \( 81 - 9 T^{2} + T^{4} \)
$43$ \( 121 + 176 T + 65 T^{2} - 2 T^{3} + T^{4} \)
$47$ \( 5476 - 148 T + 78 T^{2} + 2 T^{3} + T^{4} \)
$53$ \( 64 + 128 T + 128 T^{2} + 16 T^{3} + T^{4} \)
$59$ \( 1521 - 234 T + 45 T^{2} - 12 T^{3} + T^{4} \)
$61$ \( 1296 - 432 T + 180 T^{2} - 24 T^{3} + T^{4} \)
$67$ \( 1369 + 592 T + 113 T^{2} + 14 T^{3} + T^{4} \)
$71$ \( 1024 + 128 T^{2} + T^{4} \)
$73$ \( 3721 + 134 T^{2} + T^{4} \)
$79$ \( ( 144 + 12 T + T^{2} )^{2} \)
$83$ \( 4 + 4 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$89$ \( ( 4 + T^{2} )^{2} \)
$97$ \( 9409 + 1940 T + 303 T^{2} + 20 T^{3} + T^{4} \)
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